THE primary function of a thermal protection system (TPS) is to

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1 AIAA JOURNAL Vol., No., Feruary 1 Two-Dimensional Orthotropic Plate Analysis for an Integral Thermal Protection System Oscar Martinez, Bhavani Sankar, and Raphael Haftka University of Florida, Gainesville, Florida 311 and Max L. Blosser NASA Langley Research Center, Hampton, Virginia 381 DOI: 1.1/1.J11 This paper is concerned with homogenization of a corrugated-core sandwich panel, which is a candidate structure for integrated thermal protection systems for space vehicles. The focus is on determining the local stresses in an integrated thermal protection system panel sujected to mechanical and thermal loads. A micromechanical method is developed to homogenize the sandwich panel as an equivalent orthotropic plate. Mechanical and thermal loads are applied to the equivalent thick plate, and the resulting plate deformations were otained through a shear deformale plate theory. The two-dimensional plate deformations are used to otain local integrated thermal protection system stresses through reverse homogenization. In addition, simple eam models are used to otain local facesheet deformations and stress. The local stresses and deflections computed using the analytical method were compared with those from a detailed finite element analysis of the integrated thermal protection system. For the integrated thermal protection system examples considered in this paper, the maximum error in stresses and deflections is less than %. This was true for oth mechanical and thermal loads acting on the integrated thermal protection system. Nomenclature a = integrated thermal protection system length = integrated thermal protection system width d = height of sandwich panel (centerline to centerline) D = inverse ending stiffness matrix D = reduced ending stiffness matrix E = Young s modulus EI = equivalent flexural rigidity G = transverse shear modulus P z = pressure load acting on the two-dimensional orthotropic panel fqg = displacement vector Q ij = transformed lamina stiffness matrix Q x, Q y = shear force on unit cell Q ij = transformed lamina stiffness matrix in the nonprincipal coordinate system s = we length t BF = ottom facesheet thickness t TF = top facesheet thickness t w = we thickness T D e = deformation transformation matrix of the ith component of the corrugated core U = unit-cell strain energy v = deflection of eam w = integrated thermal protection system panel deflection Received 13 January 11; revision received 9 August 11; accepted for pulication 31 August 11. Copyright 11 y the American Institute of Aeronautics and Astronautics, Inc. Copies of this paper may e made for personal or internal use, on condition that the copier pay the $1. per-copy fee to the Copyright Clearance Center, Inc., Rosewood Drive, Danvers, MA 193; include the code 1-1/1 and $1. in correspondence with the CCC. Structural Analyst, Department of Mechanical and Aerospace Engineering; Jacos Engineering, Engineering Science Contract Group, Houston, Texas. Eaugh Professor, Department of Mechanical and Aerospace Engineering. Associate Fellow AIAA. Distinguished Professor, Department of Mechanical and Aerospace Engineering. Fellow AIAA. Aerospace Technologist, Metals and Thermal Structures Branch, Structural Concepts and Mechanics Branch. 38 y f = local y axis of facesheet y w = local y axis of we z f = local z axis of facesheet z w = local z axis of we " o = midplane strain = angle of we inclination = curvature = roots of auxiliary equation = Poisson s ratio x, y = rotations of the plate s cross section p = unit-cell length I. Introduction THE primary function of a thermal protection system (TPS) is to protect the space vehicle from extreme aerodynamic heating and to maintain the underlying structure within acceptale temperature and mechanical constraints [1]. The current state-of-the-art reusale TPS covers the outer surface of a vehicle with a layer that does not carry significant loads ut prevents the underlying loadearing structure from exceeding acceptale temperature limits. An intriguing alternative is to emed the relatively fragile insulation in a structural load-earing sandwich panel. A roust, structural facesheet would then form the outer surface of the vehicle, therey potentially removing some vehicle operational constraints, e.g., flight through rain, and reducing required maintenance. By using the insulation thickness as a structural sandwich core, there is a potential to increase structural ending stiffness, save weight, and reduce overall thickness of the vehicle wall. There are significant challenges in designing a structure to function simultaneously as an efficient aerospace load-earing structure and a thermal insulator. The challenges include understanding the key thermal-structural loads and requirements that drive the design, identifying materials that est perform in a thermal environment, and accommodating thermal expansion mismatches, thermal stresses, and effects of temperature-dependent properties inherent in these systems. Candidate concepts comprising metallic foams and innovative core materials, such as corrugated and truss cores, have een investigated. The corrugated-core sandwich panel is composed of several unit cells placed adjacent to each other. The empty space in the corrugated core will e filled with a non-load-earing insulation such as Safill. The TPS concept comines all three passive TPS concepts: heat sink,

2 388 MARTINEZ ET AL. hot structure, and insulated structure []. The sandwich structure will replace the structural skin and the insulation in the current thermal structures. Since the sandwich construction is stiffer than single skin construction, the numer of frames and stringers will e significantly reduced. Furthermore, the insulation is protected from foreign oject impact and requires less or no maintenance such as waterproofing. The integral TPS/structure [integrated thermal protection system (ITPS)] design (Fig. 1) can significantly reduce the overall weight of the vehicle as the TPS/structure performs the load-earing function. The ITPS is expected to e multifunctional (offer insulation as well as load-earing capaility). Advantages of an ITPS concept have een identified y Martinez et al. [3]. The finite element method () is commonly used to perform three-dimensional (3-D) stress analysis of the ITPS structure. However, a full-scale 3-D finite element (FE) stress analysis will e expensive and time-consuming for a quick preliminary sizing analysis of an ITPS. The same is true with design optimization, which may involve 1s of analyses. An alternative is to model the ITPS as an equivalent orthotropic plate and perform a -D plate analysis to otain the displacement and stress fields. This paper is the third in the sequel of papers written y the authors on the suject of homogenization of the ITPS. The procedures are depicted in Fig. 1a. The ITPS, which consists of repeating cells, is homogenized as an equivalent orthotropic plate, and the ABD matrices (stiffness matrices) of the equivalent plate are calculated using analytical methods. In the first paper [3], the authors descried the homogenization procedures and verified the accuracy of the extensional A, ending D, and shear C stiffness matrices y comparing with those calculated using s. In the second paper [], procedures for calculating equivalent thermal forces and moments for a given through-the-thickness temperature distriution were developed without considering shear deformations. In the current paper, the equivalent orthotropic plate is analyzed using a first-order shear deformale theory in which the transverse shear stresses are not neglected. Both pressure loads and thermal loads are considered. The plate analysis results include midplane strains f" g and curvatures fg at a given x; y. Then, a reverse homogenization procedure is used to calculate the detailed stress field from the f" g and fg otained from the plate analysis. The stresses are compared with those otained from a detailed 3-D analysis of the ITPS. It is worth pointing out that, in the present work, analytical methods are used for every step of the procedures descried aove. methods are much faster than FE analysis and are suitale when 1s of analyses have to e performed in the design optimization of the ITPS panel. methods will e extremely useful if one decides to use stochastic optimization rather than a safety-factor-ased design approach. Only a limited amount of work in micromechanical unit-cell analyses of sandwich structures is availale in the literature [ 1]; it has een summarized in our previous works [3,] and will not e repeated here. The aforementioned researchers used the forcedistortion relationship, which involved mechanics of materials equations to determine average unit-cell stresses. In the current paper, we improve on that y use of a first-order shear deformale plate theory (FSDT) [13] for the -D equivalent ITPS plate analysis, reverse homogenization, and simple eam models for the local ITPS stress analysis. The higher-order theory that includes the effects of transverse shear deformation is suitale for an ITPS ecause of the significant transverse shear loads that are carried y the corrugated core. Although such stresses can e postcomputed through 3-D elasticity equilirium equations, they are not always accurate. The classical laminate plate theory (CLPT) [1] underpredicts deflections and overpredicts uckling loads with plate length-to-thickness ratios less than. For this reason alone, it was necessary to use the Fig. 1 ITPS: a) corrugated-core sandwich panels for use as an ITPS and ) flowchart depicting procedures in homogenization of the ITPS panel and reverse homogenization to otain the detailed displacements and stresses (TFS denotes top facesheet; BFS denotes ottom facesheet).

3 MARTINEZ ET AL. 389 Fig. Dimensions of the unit cell. the in-plane forces, transverse shear forces, and ending and twisting moments; and A, C, and D are the extensional, transverse shear, and ending stiffness. The equivalent orthotropic plate is assumed to e symmetric aout its midplane, leading to a negligile extensionending coupling matrix B. The case of unsymmetric plates will e dealt with using approximate methods as discussed in this paper. The detailed homogenization procedures for determining the A, B, C, and D matrices for the corrugated-core sandwich panel were derived in Martinez et al. [1]. In the following sections, we descrie the procedures for determining the response of the ITPS to pressure and thermal loads. first-order theory in the ITPS analysis of relatively thick plates. The primary ojective and importance of using the FSDTwhen analyzing the ITPS as a two-dimensional (-D) homogenized plate was to ring out the effect of shear deformation on deflections and stresses. The paper s ojective is to derive a satisfactory approximate plate theory of an ITPS panel homogenized as a -D plate using the FSDT. The solution of the plate equations gives all necessary information for calculating stresses at any point of the thick plate. Local ITPS stresses of the facesheets and wes are derived y using the -D plate deflections w, rotations, x, y, reverse homogenization, and a transformation matrix. Reverse homogenization only captures the midplane strains and curvatures of the unit cell at the origin (see Fig. ), which resulted in a uniform stress field in the facesheets and wes. Additional local analysis had to e done through one-dimensional (1- D) eam models to capture the nonuniform stress field of the facesheet and wes when sujected to mechanical and thermal loads. The ITPS panel is sujected to a uniform pressure load and thermal edge moments, and the resulting plate deflections and local ITPS stresses are compared with a detailed FE analysis of the ITPS. II. Analysis Consider a simplified geometry of an ITPS unit cell, shown in Fig.. The z axis is in the thickness direction of the ITPS panel. The stiffer longitudinal direction is parallel to the x axis, and the y axis is in the transverse direction. The unit cell consists of two inclined wes and two thin facesheets. The unit cell is symmetric with respect to the yz plane. The upper faceplate thickness t TF can e different from the lower faceplate thickness t TB, as well as the we thickness t w. The unit cell can e identified y six geometric parameters p; d; t TF ;t BF ;t w ; (Fig. ). The equivalent stiffness of the orthotropic plate is otained y comparing the ehavior of a unit cell of the corrugated-core sandwich panel with that of an element of the idealized homogeneous orthotropic plate (Fig. 3). The in-plane extensional and shear response and out-of-plane (transverse) shear response of the orthotropic panel are governed y the following constitutive relations: 3 3( ) N A 3 3 "o Q 3 C 3 or ffgkfqg (1) M 3 3 D In Eq. (1), " o and are the in-plane and transverse shear strains, and are the ending and twisting curvatures; fng, fqg, and fmg, are A. Uniform Pressure Loading A simply supported orthotropic sandwich panel of width (y direction) and length a (x direction) was considered for a -D plate analysis (Fig. 3). The simply supported oundary conditions are descried as w;y; wa; y M x ;y M x a; y wx; ; wx; M y x; ; M y x; () The panel is exposed to a uniform pressure load that can e represented y the doule Fourier sine series, as shown in Eq. (3): P Z x; y X1 X 1 mx ny P mn sin sin (3) a m1 n1 In the aove equation, P mn 1P o = mn for uniform pressure loads [13], where P o is the intensity of the uniformly distriuted load. The ITPS panel is also assumed to have the following deformations that satisfy the simply supported oundary conditions: wx; y X1 X 1 mx ny A mn sin sin (a) a xx; y X1 yx; y X1 m1 n1 X 1 m1 n1 X 1 m1 n1 mx B mn cos a sin ny mx ny C mn sin cos a () (c) In the aove equations, wx; y is the out-of-plane displacement, and x x; y and y x; y are the plate rotations. The unknown constants A mn, B mn, and C mn were otained y sustituting the constitutive relations in the form of the assumed deformations from Appendix A [Eqs. (A) and (A3)] into the differential equation of equilirium [Eq. (A1)]. This results in a system of three linear equations for the unknown coefficients as shown: L P o A I B II C L 1 III D z y Fig. 3 Equivalent orthotropic thick plate for the unit cell of the corrugated-core sandwich panel. Fig. Half of the top face under the action of a uniform pressure loading.

4 39 MARTINEZ ET AL. m D 11 a n D mn kc D 1 D a mn n D 1 D D a m D a kc kc kc m a kc n m kc a n kc m a kc n 3 ( Bmn C mn A mn ) ( ) P mn () After solving for the unknown constants, the deflections were otained at a given x; y coordinate on the -D orthotropic sandwich panel using Eq. (). The results in the aove series converged for m n 3, resulting in a total of 9 terms in Eq. () [1]. It should e mentioned that the shear correction factor k in Eq. () was assumed to e unity, as C and C were directly predicted using micromechanics [3]. B. Top Facesheet Local Deflection The top facesheet was directly exposed to a uniform pressure loading. The uniform pressure loading that acted on the thin facesheet resulted in local deflections that were not captured y the -D FSDT. The FSDT analysis resulted in midplane strains and curvatures at the center of the unit cell. The facesheet local deformation vector due to the pressure requires additional analysis. A 1-D eam prolem was used to calculate the additional local deformation of the facesheet under uniform pressure load. Because of symmetry, only half of the top facesheet was analyzed (Fig. ). A matrix structural analysis method (FE) was used to determine the deflection of the top facesheet. Half of the top facesheet was modeled using two eam elements. Displacement and rotation oundary conditions were imposed on the two ends of the facesheet. A torsional spring element with a rotation oundary condition was used at the middle node to capture the resistance to rotation offered y the we connected to the facesheet. The elements are indicated y roman numerals in Fig.. Because of symmetry, A and c were zero. Because of the displacement constraint of the we and the facesheet, the deflection at junction B was set to zero, i.e., v B. The clamped oundary condition at node D resulted in D eing zero. The consistent, or work, equivalent loads on each element due to the uniform pressure loading were computed using the eam element shape functions [1] as shown elow: 8 >< >: F i C i F j C j 9 >= >; Z L P o 8 >< y f y f >: 1 3y f y3 f L L 3 y3 f L L 3y f y3 f L L 3 y f L y3 f L 9 >= dy f () In Eq. (), i and j correspond to the left and right nodes of the eam and torsional elements, and L corresponds to the length of the eam element. The element stiffness matrix for the eam and torsional spring elements were assemled accordingly to otain the gloal stiffness matrix and gloal force vector: EI 1 L L 3 L 1 8 >< >: L 1 L P ol 1 P ol P o L 1 P ol 1 1 L 1 L 1 1 L 1 L 3 9 >= >; 3 8 >< >: v A v C B 9 >= >; >; KfqgfFg () In Eq. (), L 3 was the length of the we. Equation () was solved to otain the nodal displacements, which were sustituted ack in the interpolation functions to otain the transverse deflection of the facesheet: v I y f 1 3y f L y3 f v L 3 A y f L y3 f L B (8) v II y f y y f L y3 f 3y f L B L y3 f v L 3 C (9) The 1-D deflections from Eqs. (8) and (9) were superimposed with the plate deflection from Eq. (a). C. Stresses due to Pressure Loading 1. Stress Because of Midplane Strain and Curvature The FSDT analysis of the equivalent plate yields midplane strains, curvatures, and transverse shear strains at given x; y of the equivalent plate. These deformations are called the macrodeformations [3], and they are the deformations in the equivalent orthotropic plate. To calculate the stresses in the ITPS, we apply the aforementioned macrodeformations to a unit cell situated at the same point. Then, using the transformation matrices defined y Martinez et al. [3], the (actual) deformations in the facesheets and the we were calculated. From the constitutive relations of the facesheet and we materials (layup in the cases of laminates), the stresses are calculated. A flowchart (Fig. ) outlines the stress calculation procedure. In the flowchart, the x and y locations correspond to the ITPS -D equivalent plate, and y and z correspond the local location on either the face or we; Fig. a. If e in Fig. was equal to 3 or (left or right we), then the micro-mid-plane strains and curvatures were a Fig. Local stress flowchart for an ITPS as a -D plate with transverse shear force effects consideration.

5 MARTINEZ ET AL. 391 Fig. Unit cells: a) ITPS unit cell under the action of transverse shear force, and ) half-itps unit cell. function of y w, which would result in we stresses as a function of y w. There will e additional stresses in the top facesheet that result from the local effects of the pressure loading. These local stresses are determined from the local deflections derived in Eqs. (8) and (9) and added to that resulting from the equivalent plate analysis in Eq. (). The stresses calculated from Eqs. (8) and (9) neglect a shear deformation term ecause the facesheets and wes are thin when considered y themselves and compared with the unit-cell dimensions.. Stress due to Transverse Shear Strains In this section, we descrie the procedures to calculate the stresses in the facesheets and we due to transverse shear force Q y acting on the equivalent orthotropic plate. Figure illustrates the ITPS halfunit cell under the action of various forces that resulted from a transverse shearing force Q y. The unknown forces P, R, and F for unit Q y were determined from Castigliano s second theorem [1]. Then, using the procedure descried y Martinez et al. [1], the ending moments at a cross section of the facesheets or wes were determined (Appendix B). From the moment equations (Appendix B), the resulting curvature in either of the facesheets or wes was determined: eq y D e 1 y y; x; y D e 11 D e M ij yq y x; y (1) The flowchart shown in Fig. takes into account the moments from Eq. (A1) and the transverse shear force from Eq. (A3) for local stress determination. The superscript e in Eq. (1) refers to a component assignment of the ITPS. An e of 1,, 3, and refers to the top facesheet, ottom facesheet, left we, and right we. It should e mentioned that, in applying Castigliano s theorem, we accounted only for the strain energy due to ending in various segments of the cross sections. The energy due to in-plane forces and transverse shear force was assumed to e negligile compared with the ending energy since the thickness of the facesheet and wes are small when compared with its length. 3. Local Stress due to the Uniform Pressure Loading: One-Dimensional Analysis The calculation of local stresses in the top facesheet due to pressure loading is a continuation of the analysis descried aove that determines the local transverse displacements [see Eqs. (8) and (9)]. The free ody diagram of the top facesheet under the action of a uniform pressure loading is shown in Fig.. In Fig. 8, the moments at points A and B were determined y multiplying the element stiffness matrix with the corresponding node displacement vector; see Appendix C. The stresses at any local point of the top facesheet were determined y multiplying the moment vector with the inverse of the ending stiffness matrix and the lamina stiffness matrix of the facesheet as shown elow. A superscript value of 1 refers to the top facesheet: 3 xx y f ; z f yy y f ; z f 1 yy y f ; z f z f Q 1 11 Q 1 1 Q 1 1 Q 1 1 Q 1 Q 1 Q 1 1 Q 1 Q 1 3 D 1 11 D 1 1 D 1 D 1 1 D 1 1 D 1 D 1 1 D 1 D D 1 1 D 1 M ij y M ij y 3 (11) D. Unsymmetric Integrated Thermal Protection System Panel Configuration The methods descried so far are for an ITPS with symmetric facesheet configurations with respect to the unit-cell y axis, i.e., B matrix is zero. However, certain designs may call for different materials or thicknesses for the top and ottom facesheets that make it perform at its optimum for mechanical and thermal applications. Ceramic matrix composites and high-temperature metals are possile choices for the top facesheet due to their high service temperatures. Aluminum alloys and composite materials are of preference for the ottom facesheet due to their high specific heats and mass-efficient structural properties. An ITPS with different materials for the top and ottom facesheets results in a stiffness matrix with a nonzero coupling stiffness matrix, B. This type of ITPS configuration will e referred to as an unsymmetric ITPS. The presence of the ending-extensional coupling stiffness matrix causes considerale difficulty in otaining solutions to a plate prolem using either the FSDT or CLPT method. As an alternative, the reduced stiffness matrix approach [1], wherein B is assumed to e equal to zero and D is modified, could e used. Then, the techniques descried aove for analysis of orthotropic plates can e directly used. The reduced stiffness matrix [Eq. (1)] was used in the plate solution of an orthotropic plate with a uniform pressure load. The use of the reduced stiffness matrix tends to reduce the effective stiffness P o y f y f P L + y ) o( f ( L + y f ) M C M AB C M C B C M BC L y f y f F B Fig. Free ody diagram of section BC (see Fig. ) of the top facesheet. Fig. 8 Free ody diagram of section AB (see Fig. ) of the top facesheet.

6 39 MARTINEZ ET AL. of the plate, which results in increased ending deflections while uckling loads are decreased when compared with equivalent symmetric orthotropic plates. The corresponding error in the out-ofplane plate displacement and local stresses was investigated and compared with the FE results. The reduced ending stiffness matrix D is given y [1]: D DBA 1 B (1) E. Thermal Stress Moment Resultants The ITPS panel is exposed to extreme reentry temperatures that result in various temperature distriutions through the ITPS thickness as a function of reentry time (transient heat transfer) [1]. The temperature distriution causes the ITPS to deform due to the resulting thermal force and moments that were determined y Martinez et. al [18]. An analytical solution was derived to predict the thermal deflection of the ITPS panel when sujected to a throughthe-thickness temperature distriution. A -D plate analysis was needed to determine the response of the ITPS when it was sujected to thermal moments. Consider a rectangular plate simply supported along the nonloaded edges and ent y moments distriuted along the edges x ;a (Fig. 9). The distriuted moment in Fig. 9 is equal to the previously calculated thermal moment from Martinez et al. [18]. Typical ranges of L=h for the ITPS will e from 1 to, which result in nonnegligile shear effects. An FSDTapproach was implemented for the deflection solution of the ITPS plate. The oundary conditions for the thermal plate prolem are as follows: w ; x y ; w ; y x ;a Mx T D 11 x;x x ;a (13) The deflections and rotations are assumed such that they satisfy all displacement oundary conditions along the edges y and y : We assume solutions to e of the form of an exponential series function; see Eq. (D). Sustituting the assumed solution from Eq. (D) in the differential equations [Eq. (D1)] yields a set of homogeneous linear equations for the unknown coefficients,, and as shown elow: D 11 D A D 1 D A 3 det D 1 D D D A A A A A A 89 < = 89 < = : ; : ; (1) The lamdas were solved y setting the determinant of the coefficient matrix to zero. The solution to Eq. (1) yielded six solutions ;. The six roots yielded six terms in Eq. (D), which resulted in three equations with six unknowns per equation [Eqs. (D3 D)]. Eighteen unknown constants resulted from the equations in Appendix D, which were not possile to solve with only six oundary conditions. However, a relation was made etween all 18 constants (see Appendix D), which reduced the numer of independent constants from 18 to. A unique solution to each unknown constant was now availale ecause of the equal numer of unknown constants and oundary conditions. The moment distriution along the edges of the plate was represented y the Fourier sine series as M x XN n1 M n sin nx a (1) For the case of a uniform distriution of the ending moments, the term M n was represented as M n M T x =n. A system of six linear equations was otained y sustituting Eq. (D) into Eq. (1) and Eq. (1) into Eq. (13): 8 >< >: >= >; e 1 e e 3 e e e C 1 C C 3 C C C C 1 e 1 C e C 3 e 3 C e C e C e D D 1 1 D D D D 1 3 D D 1 D D 1 D D 1 D D 1 1 e 1 D D 1 1 e 1 D D 1 1 e 1 D D 1 1 e 1 D D 1 1 e 1 D D 1 1 e >< >: M T y m My T m 9 >= >; (1) wx; y XN n1 A n x sin ny yx; y XN n1 xx; y XN C n x cos ny n1 B n x sin ny (1) By solving the set of six linear equations, the assumed deflection equations [Eq. (1)] were solved. Equations (D) and (D) were used to otain the solutions to the plate rotation in the x and y directions. The same procedure was applicale for uniformly distriuted The deflection and curvature solutions of the plate are taken in the form of a series where A n x, B n x, and C n x are all unknown functions of x only. It was assumed that the two unloaded edges are simply supported; hence, each term of the series satisfies the oundary conditions w and x on these two sides. It remains to determine A n x, B n x, and C n x in such a form to satisfy the oundary conditions on the sides x and x a, and the equation of equilirium [Eq. (A1)]. Taking the assumed deflection and rotations in the form of a series and sustituting them into the equilirium equations results in three ordinary differential equations [Eq. (D1)]. h x L T M x Fig. 9 ITPS orthotropic plate sujected to uniformly distriuted thermal end moments. y a

7 393 MARTINEZ ET AL. Fig. 1 ITPS out-of-plane deformation. FE a) model and ) mesh. III. Results A. Integrated Thermal Protection System Out-of-Plane Displacement: Pressure Load For verification of the accuracy in prediction of the deflection and stress results of an ITPS sandwich panel under the action of a uniform pressure loading, an ITPS panel with the following dimensions was analyzed: p mm, d 1 mm, ttf 1: mm, tbf 1: mm, tw 1: mm,, a 1 m, and 1 m. An Inconel 1 alloy was used as an example to verify the analytical models for determining local stresses in the ITPS sandwich panel (E GPa, :8). A 1-unit-cell ITPS sandwich panel was modeled for the analysis (Fig. 1a). Because of symmetry of the ITPS panel, only a quarter of the plate was modeled. A 3-D FE analysis was conducted on the ITPS plate using the commercial ABAQUSTM version. FE program. Eight node shell elements were used to model the facesheets and wes of the unit cell. The shell elements have the capaility to include multiple layers of different material properties and thicknesses that are needed to model a laminate. Three integration points were used through the thickness of the shell elements. The model consisted of 1,8 nodes and 1 elements (Fig. 1). The ITPS panel was sujected to a uniform pressure loading of 8,9 Pa (1 psi), and symmetric oundary conditions were imposed at the edges of the ITPS plate. The ottom facesheet had an out-of-plane displacement constraint w ; y w x;, while the top facesheet only had a rotation constraint in all three directions along x and y (Fig. 11). The FE model was constrained with the appropriate oundary conditions from Eq. () and appropriate symmetric oundary conditions. The ysymmetric oundary conditions are v x; = Rx x; = Rz x; = and the x-symmetric oundary conditions are potentially have an unlimited numer of fictitious unit cells depending on what x and y points are chosen. Since the ITPS model contained 1 unit cells, for comparison, the homogenized ITPS -D plate was divided into a 1 1 grid. The accuracy was verified through FE results and a convergence study. Each intersection of the grid lines corresponded to an analytical ITPS unit cell that corresponded to the center location of the unit cell. The curvatures in the x and y directions at each analytical grid intersection were assumed to e constant for the entire unit cell; however, the curvatures in the x and y directions from the 3-D results were not constant within each unit cell. The curvatures varied through each unit cell. This stress gradient effect could not e captured y the analytical homogenized plate model. The out-of-plane displacement contour of the ITPS is illustrated in Fig. 1. The out-of-plane displacement results at x a= were compared with Out-of-Plane Displacement (mm/po) ending moments along the y axis. In the case of simultaneous action of couples along the entire oundary of the ITPS plate, the deflections and moments can e otained y suitale superposition of the results otained from the detailed discussion. x 1-8 FSDT CLPT Length Along x=a/ (m) Out-of-Plane Displacement (mm/po) Fig. 1 x 1-9 FSDT CLPT Length Along x=a/ (m) Fig. 13 Out-of-plane displacement comparison etween and analytical solution of the top and ottom facesheets. Out-of-Plane Displacement (mm/po) The out-of-plane displacements of the ITPS panel at x a= were extracted and compared with the results otained from Eqs. (a), (8), and (9). The homogenized ITPS -D plate can Out-of-Plane Displacement (mm/po) u a=; Rx a=; Rz a=; Fig. 11 FE oundary conditions for the ITPS plate. x Length Along x = a / (m) x Length Along x = a / (m) Fig. 1 Out-of-plane displacement comparison etween and analytical solution of the top and ottom facesheets for an unsymmetric ITPS.

8 39 MARTINEZ ET AL. the analytical results (Fig. 13). As seen in Fig. 13, the analytical outof-plane displacement results are within a 1 % difference from the results. There was greater percentage difference near the oundary of the ITPS ecause the oundary conditions created localized effects that were not captured y the FSDT method and the 1-D eam analysis. At the center of the ITPS plate, the structure acted more like the homogenized -D plate ecause it was farther away from the oundary. The superposition of the displacement results otained from the FSDT method and the 1-D eam analysis resulted in a less than % difference and an accurate representation of the top facesheet deflection response when sujected to a uniform pressure load. The ottom facesheet FE displacement results agreed well with the displacement results otained from the FSDT method. The ottom facesheet deflection acted more like a -D plate ecause of the asence of local effects from the uniform pressure loading. To verify the applicaility of present methods to unsymmetric ITPS, an ITPS with a different ottom facesheet was considered. The ottom facesheet was assumed to e of an aluminum alloy (E :19 GPa, 1 :3). An ITPS panel with the same dimensions as mentioned aove was modeled and analyzed for investigation of the error in using the reduced stiffness matrix in the FSDT method for an unsymmetric ITPS panel. The local stresses were extracted at x a= and compared with the analytical stresses that resulted from the FSDT method y use of the reduced stiffness matrix. From Fig. 1, it can e noted that the plate displacement results were not affected y using the reduced stiffness matrix in the FSDT plate analysis. The largest difference etween the and analytical results was less than %. Normalized Core Thickness 1 Tale 1 Thermal moments of Inconel 1 under the s reentry temperature distriution Stiffness M x, Nm=m M y, Nm=m 1:E 8:8E FE 1:1E 9:E % diff..1. s -1 8 Temperature (K) Fig. 1 ITPS temperature distriution at s reentry time. Boundary condition 1 Boundary condition Boundary condition 3 Tale Boundary conditions of the ITPS panel Top face rotation Top face displacement Bottom face rotation Bottom face displacement X X X X B. Integrated Thermal Protection System Out-of-Plane Displacement: Temperature Distriution From the results otained y Martinez [19] on the analysis of a plate under the action of thermal edge moments, confidence was gained with the analytical procedure for a plate solution of an orthotropic thick plate under the action of uniformly distriuted edge moments. The next step was to apply that approach to analytically predict the out-of-plane displacement of an ITPS under the action of thermal moments from an applied temperature distriution. The temperature distriution that was considered for the analysis was at the s reentry time of a space-shuttle-type vehicle, as descried in [19] (Fig. 1). The ITPS was composed of Inconel 1, and the resulting thermal moments for Inconel 1 are calculated and compared with the FE verification procedure discussed y Martinez [19]. The greatest percentage difference etween the analytical and comparison was less than % for M T x. The thermal moment results (Tale 1) were used to determine the analytical out-of-plane displacement of the ITPS panel. Tale 1 consists of two moments. The aforementioned thermal plate analysis with edge moments on the x and y axes can e used through suitale superposition (see Fig. 1). The same ITPS panel that was modeled for the pressure load example was used for the temperature distriution example, except for this case, the thickness of the faces and wes was increased from 1. mm (.9 in.) to. mm (.8 in.). The numer of nodes and elements remained the same, as well as the oundary conditions. The s temperature distriution was included in the input file, and the out-of-plane displacement of the ITPS was extracted at x a= after analysis. There was no need to do any local out-ofplane displacement prolem for this case ecause the temperatures of the top and ottom facesheets were considered to e constant ecause of the thin faces when compared with the ITPS thickness. Therefore, the faces were only under the action of a thermal expansion. The out-of-plane displacement result of the ITPS panel under the action of the fourth-order temperature distriution is shown in Fig. 1 along with the CLPT [19] and FSDTanalytical results. The FSDT method for predicting out-of-plane displacement for an ITPS under the action of uniformly distriuted edge moments does not predict accurate results for this particular ITPS panel. The FSDT method overpredicts the actual FE displacement y %, and the CLPT [] method underpredicts the actual FE displacement y X X X ITPS Out-of-Plane Displacement (mm) 8 CLPT FSDT TFS- BFS- Fig Length Along x = a / (m) a) ) ITPS a) out-of-plane thermal displacement and ) thermal displacement contour (TFS denotes top facesheet; BFS denotes ottom facesheet).

9 MARTINEZ ET AL. 39 ITPS Out-of-Plane Displacement (mm) 3 1 TF1 CLPT BF1 TF BF TF3 BF3 FSDT Length Along x = a / (m) Fig. 1 ITPS out-of-plane displacement due to a fourth-order temperature distriution for various edge oundary conditions. ITPS Center Panel Displacement (mm) 1 1 FSDT % difference 1 1 Length / Height (L / h) Fig. 18 ITPS center panel displacement for various L=h values. %. The edge oundary conditions of the top and ottom facesheets of the model were modified in hopes of otaining a good comparison etween the and analytical results. The set of oundary conditions are listed in Tale. An X represents a constrained condition. The ITPS out-of-plane displacement results for all different oundary conditions are shown in Fig. 1 and compared with the CLPT and FSDT results. As seen from Fig. 1, the oundary condition does not affect the center plate out-of-plane displacement. The different oundary conditions had an effect on the out-of-plane displacements that were near the edge oundary. The different oundary conditions cause different oundary effects that affect the ITPS out-of-plane displacement that cannot e captured y the FSDT or CLPT methods. The change in oundary condition did not lead to a etter comparison etween the and analytical results Percentage Difference % σ xx (MPa/Po) σ yy (MPa/Po) x Length at x = a / (m) x Length at x = a / (m) Fig. Top facesheet stress in the x and y directions of the ITPS panel. The thermal moments that resulted from the fourth-order temperature distriution caused the ITPS panel to end due to thermal stresses. Since the ITPS plate had a low L=h value, the thermal moments acted on a short edge length of the ITPS, which introduced other local oundary effects into the results. These local ITPS plate effects that resulted from the thermal moments were not fully captured y the FSDT and CLPT methods when the plate had a low L=h ratio. The L=h ratio of the ITPS was increased to investigate what L=h ratio of an ITPS will result in a less than a % prediction error of the and analytical out-of-plane thermal displacement results. The L=h ratio in the model was increased y adding more unit cells to the ITPS plate while keeping the ITPS thickness constant. For ITPS plates with large L=h ratios, the thermal moments now acted on a long edge length, and all the local ITPS oundary effects were minimized. The ITPS panel was ale to end as a plate, and the ending ehavior was captured y the FSDT that was outlined aove. The resulting maximum out-of-plane displacement was extracted from the output after analysis and compared with the maximum analytical plate displacement for various L=h ratios. The percentage error decreased from % for low L=h ratios to % for high L=h ratios (Fig. 18). The analytical and results were in good agreement for that particular L=h ratio, which resulted in a less than % prediction error of the ITPS out-of-plane displacement from a fourth-order temperature distriution (see Fig. 19). The and analytical stress results were plotted at x a= for an ITPS panel with an L=h 18 (Figs. and 1). C. Integrated Thermal Protection System Local Stress The local stresses in the x and y directions were extracted from the output after analysis at x a=. The analytical top facesheet stresses were computed y superimposing the stresses results from Fig. and Eq. (1). The ottom facesheet and we stresses were computed from Fig.. All stress results were computed at z f or w t=, which was the ottom surface of each component. Similar to the out-of-plane displacement results, the percentage difference etween the and analytical results was greater at the oundary of the ITPS; see Figs.. The percentage difference near the oundary of the ITPS was 9%. The percentage difference ITPS Out-of-Plane Displacement (mm) 1 BF- TF- FSDT....8 Length Along x = a / (m) Fig. 19 ITPS out-of-plane displacement due to a temperature distriution for an ITPS with L=h 18. σ xx (MPa/Po) σ yy (MPa/Po) 1 x Length Along x = a / (m) x Length Along x = a / (m) Fig. 1 Bottom facesheet stress in the x and y directions of the ITPS panel.

10 39 MARTINEZ ET AL. Fig. Top facesheet stress in the x and y directions of an unsymmetric ITPS panel. σ xx (MPa/Po) σ yy (MPa/Po) x Length Along x = a / (m) 1 x Length Along x = a / (m) Fig. 3 Bottom facesheet stress in the x and y directions of an unsymmetric ITPS panel. was less than % for stress results near the center of the ITPS plate. The percentage difference for the we stress in the x direction was less than 1% when compared with the results. The percentage difference for the we stresses in the y direction was less than 3% at the center of the plate and greater than % near the oundary of the ITPS panel. The analytical local stress of the right we in the y w direction was different than the results ecause, near the oundary of the ITPS, there was a large change in curvature due to the local oundary effect that was not captured y the analytical model. The curvature was changing through the length of the unit cell while, in the analytical model, the curvature was kept constant. The local stress prediction gets etter when the local stresses are compared at the center of the plate. There are less oundary effects, and the curvature can e assumed constant for a unit cell. For investigation of the penalty of using the reduced stiffness matrix in the FSDT method for an unsymmetric ITPS panel, an unsymmetric panel with the same dimensions as mentioned aove was modeled and analyzed. The difference in this new model was that the top facesheet and wes were composed of Inconel 1, and the ottom facesheet was composed of an aluminum alloy (E :19 GPa, 1 :3). The local stresses were extracted at x a= and compared with the analytical stresses that resulted from the FSDT method y use of the reduced stiffness matrix. The comparison etween the FE results and the stress results otained from the FSDT method are illustrated in Figs. and 3. The results indicated that there was a % difference etween the analytical and FE results in the top facesheet when using the reduced stiffness matrix. The use of the reduced stiffness matrix in the FSDT method did not have any effect on the ottom facesheet and we stresses when compared with the results. The difference was much more noticeale in the top facesheet ecause of the local effects that resulted from the uniform pressure load. IV. Conclusions A -D plate solution for determining out-of-plane displacement of an orthotropic thick plate was estalished and verified with for a uniform pressure load and uniform thermal edge moments. Additional local facesheet deformations and stress were superimposed with the -D plate results to capture the local effects of the distriuted pressure load on the facesheets. Kirchhoff hypotheses and Euler Bernoulli plate equations were used for the local displacement and stress analysis of the facesheets and wes. The symmetric and unsymmetric plates resulted in a less than % prediction error in the plate s out-of-plane displacements when compared with the. The plate solution provided less than % prediction error results for ITPS panels with L=h ratios greater than 18. An ITPS panel with a low L=h introduced local oundary effects that contriuted to the panel s out-of-plane displacement. A 1-D eam analysis was done on the top facesheet to account for the local ending of the top facesheet due to its thin thickness. The local oundary effects that introduced local displacements and stresses were not fully captured with either the FSDT method or the CLPT method. Appendix A: Plate Equilirium Equations and Constitutive Relations The plate equilirium equations x Q x P z The plate constitutive relations are ) ( Mx M y M xy Qy D 11 D 1 D 1 D D Q xy Q y k C C 3( x;x y;y x;y y;x y w ;y x w ;x ) (A1) (A) (A3) Appendix B: Moments in the Half-Unit Cell In this Appendix, we provide the expressions for ending moments in various parts of the unit cell due to unit transverse force [see Eq. (1)]. The detailed derivations can e found in []: M AB yf y y f (B1) M BC ypy y p f (B) M DE y1 Fy P Rp Jd y p f (B3) M EG yry y f (B) M BE yff y cos Ppf y cos Hy sin (B) y s Appendix C: Local Bending Stress in Facesheets due to a Uniform Pressure Loading The moments in the equations shown elow are the result of the uniform pressure loading acting on the top facesheet. The moments were otained from a 1-D FE analysis. The moments were multiplied

11 MARTINEZ ET AL. 39 with the ending stiffness matrix of the facesheet to otain facesheet stress: M C EI L 3 L B L v B (C1) k i i 13 k i 1 ki 3 ki 11 i k i ki 11 D i i ki 1 ki 1 k i 11 D 11 i D C (D) (D8) M A EI L 3 L 1 v A L 1 B 1 (C) k i D i D C (D9) The ending moment distriution in the top facesheet consists of two equations: k i 1 ki 1 D 1 D i (D1) M BC y ym C P o y y p f (C3) k i 3 C (D11) M AB y yf B y M C P o L y y f (C) k i 13 C i (D1) Appendix D: Edge Moments In this section, the assumed solutions to the plate deflection with edge moments are provided. A relationship is also made to reduce the numer of unknown constants from 18 to. Ordinary differential equations to a -D plate with edge moments: D 11 B nxd C B n xd 1 D Cnx C A nx D CnxD C C n xd 1 D B nx C A n x C A nxc A n xc B nxc C n x A n x XN i1 B n x XN i1 C n x XN i1 i e ix i e ix i e ix (D1a) (D1) (D1c) (Da) (D) (Dc) A n x 1 e 1x e x 3 e 3x e x e x e x B n x 1 e 1x e x 3 e 3x e x e x e x C n x 1 e 1x e x 3 e 3x e x e x e x k i i 1 k i 3 ki ki 13 i k i ki 11 C i i ki 1 ki 1 (D3) (D) (D) (D) Acknowledgments This research is sponsored y a NASA grant under the Constellation University Institutes Project. The program manager is Claudia Mayer at NASA John H. Glenn Research Center at Lewis Field. References [1] Zhu, H., Design of Metallic Foams as Insulation in Thermal Protection Systems, Ph.D. Dissertation, Univ. of Florida, Gainesville, FL,. [] Blosser, M. L., Advanced Metallic Thermal Protection Systems for Reusale Launch Vehicles, Ph.D. Dissertation, Univ. of Virginia, Charlottesville, VA,. [3] Martinez, O., Sankar, B. V., Haftka, R., Blosser, M., and Bapanapalli, S. K., Micromechanical Analysis of a Composite Corrugated-Core Sandwich Panel for Integral Thermal Protection Systems, AIAA Journal, Vol., No. 9,, pp doi:1.1/1.9 [] Martinez, O., Sharma, A., Sankar, B. V., Haftka, R., and Blosser, M. L., Thermal Response Analysis of an Integrated Thermal Protection System for Future Space Vehicles, AIAA Journal, Vol. 8, No. 1, 1, pp doi:1.1/1.8 [] Fung, T. C., Tan, K. H., and Lok, T. S., Analysis of C-Core Sandwich Plate Decking, Proceedings of the 3rd International Offshore and Polar Engineering Conference, Mountain View, CA, Vol., 1993, pp. 9. [] Fung, T. C., Tan, K. H., and Lok, T. S., Elastic Constants for Z-Core Sandwich Panels, Journal of Structural Engineering, Vol. 1, No. 1, 199, pp doi:1.11/(asce)33-9(199)1:1(3) [] Liove, C., and Huka, R. E., Elastic Constants for Corrugated Core Sandwich Plates, NACA TN-89, 191. [8] Lok, T. S., Cheng, Q., and Heng, L., Equivalent Stiffness Parameters of Truss-Core Sandwich Panel, Proceedings of the Ninth International Offshore and Polar Engineering Conference, Vol., International Soc. of Offshore and Polar Engineers, Brest, France, 1999, pp [9] Valdevit, L., Hutchinson, J. W., and Evans, A. G., Structurally Optimized Sandwich Panels with Prismatic Cores, International Journal of Solids and Structures, Vol. 1, May, pp doi:1.11/j.ijsolstr... [1] Lok, T. S., and Cheng, Q., Elastic Stiffness Properties and Behavior of Truss-Core Sandwich Panel, Journal of Structural Engineering, Vol. 1, No., May, pp. 9. doi:1.11/(asce)33-9()1:() [11] Fung, T. C., Tan, K. H., and Lok, T. S., Shear Stiffness DQy for C-Core Sandwich Panels, Journal of Structural Engineering, Vol. 1, No. 8, 199, pp doi:1.11/(asce)33-9(199)1:8(98) [1] Liove, C., and Batdorf, S. B., A General Small Deflection Theory for Flat Sandwich Plates, NACA TN-1, 198. [13] Whitney, M., Structural Analysis of Laminated Anisotropic Plates, Technomic, Lancaster, PA, 198, Chaps. 9.

12 398 MARTINEZ ET AL. [1] Reddy, J. N., Mechanics of Laminated Composite Plates Theory and Analysis, CRC Press, Boca Raton, 199, Chaps.. [1] Martinez, O., Bapanapalli, S., Sankar, B. V., Haftka, R., and Blosser, M., Micromechanical Analysis of a Composite Truss Core Sandwich Panel for Integral Thermal Protection Systems, th AIAA/ASME/ ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Newport, RI, AIAA Paper -18, May. [1] Sankar, B. V., and Kim, N. H., Introduction to Finite Element Analysis and Design, Wiley, NY, 9, Chaps. 3. [1] Bapanapalli, S. K., Martinez, O., Sankar, B. V., Haftka, R. T., and Blosser, M. L., Analysis and Design of Corrugated-Core Sandwich Panels for Thermal Protection Systems of Space Vehicles, th AIAA/ ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Newport, RI, AIAA Paper -19, May. [18] Martinez, O., Sankar, B. V., and Haftka, R. T., Thermal Response Analysis of an Integral Thermal Protection System for Future Space Vehicles, ASME International Mechanical Engineering Congress and Exposition, Chicago, IL, American Soc. of Mechanical Engineers Paper -1, Fairfield, NJ, Nov.. [19] Martinez, O., Micromechanical Analysis and Design of an Integrated Thermal Protection System for Future Space Vehicles, Ph.D. Dissertation, Univ. of Florida, Gainesville, FL,. [] Timoshenko, S., and Woinowsky-Kreiger, S., Theory of Plates and Shells, Timoshenko, McGraw Hill, New York, 199, pp R. Ohayon Associate Editor